You would? I did all my JAA ratings in Europe and then converted them to FAA. I spent way too much money. The other way around is so much cheaper. I understand the hassle though. Converting a license is always a hassle, but think about the $30,000+ you could be saving by doing a little bit of research and finding the right way of converting. Seems to me it's worth it.
Good luck at Bristol, have heard good things.